Nil-prime ideals of a commutative ring

TL;DR

This paper introduces and explores nil-prime ideals in commutative rings, generalizing prime ideals by incorporating nilpotent elements, and studies related nil-versions of classical algebraic concepts.

Contribution

It defines nil-prime ideals and develops the theory of nil-versions of algebraic concepts like maximal, minimal, and principal ideals in commutative rings.

Findings

Characterization of nil-prime ideals

Introduction of nil-maximal, nil-minimal, and nil-principal ideals

Extension of classical results to nil-versions

Abstract

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P of R is a nil-prime ideal if there exists x \in N(R) and whenever ab \in P, then a \in P or b \in P or a+x \in P or b+x \in P for each a,b \in R. Also, we introduce nil versions of some algebraic concepts in ring theory such as nil-maximal ideal, nil-minimal ideal, nil-principal ideal and investigate some nil-version of a well-known results about them.